The affine Hecke category, defined using affine Soergel bimodules, categorifies the affine Hecke algebra. I will compare the derived horizontal trace of the affine Hecke category with the elliptic Hall algebra, and with the derived category of the commuting stack. In particular, I will describe certain explicit generators for the trace category and some categorical commutation relations between these. This is a joint work with Andrei Negut.

In 2009, Kahn and Markovic proved the Surface Subgroup Theorem, and they constructed a ubiquitous collection of \pi_1-injective immersed surfaces in closed hyperbolic 3-manifolds. Hamenstadt later showed that any cocompact lattice of some simple rank 1 Lie group other than SO(2m,1) (m >= 1) has a surface subgroup. Recently, we constructed \pi_1-injective immersed surfaces in closed hyperbolic 2n-manifolds, which addresses the cases missing from Hamenstadt’s work. This is joint work with Jeremy Kahn.